Some improvements for a class of the Caffarelli-Kohn-Nirenberg inequalities

Abstract

In this paper, we concern a weighted version of the Hardy inequality, which is a special case of the more general Caffarelli-Kohn-Nirenberg inequalities. We improve the inequality on the whole space or on a bounded domain by adding various remainder terms. On the whole space, we show the existence of a remainder term which has the form of ratio of two weighted integrals. Also we give a simple derivation of the remainder term involving a distance from the manifold of the “virtual extremals”. Finally, on a bounded domain, we prove the existence of remainder terms involving the gradient of functions.